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Báo cáo hóa học: " Research Article Eigenvalue Problems and Bifurcation of Nonhomogeneous Semilinear Elliptic Equations in Exterior Strip Domains"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Eigenvalue Problems and Bifurcation of Nonhomogeneous Semilinear Elliptic Equations in Exterior Strip Domains | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 14731 25 pages doi 10.1155 2007 14731 Research Article Eigenvalue Problems and Bifurcation of Nonhomogeneous Semilinear Elliptic Equations in Exterior Strip Domains Tsing-San Hsu Received 19 July 2006 Revised 10 October 2006 Accepted 20 October 2006 Recommended by Patrick J. Rabier We consider the following eigenvalue problems -Aw u X f u h x in o u 0 in o u G H0 O where X 0 N m n 2 n 1 0 G M c Rm is a smooth bounded domain S M X Rn D is a smooth bounded domain in RN such that D GG S o S D. Under some suitable conditions on f and h we show that there exists a positive constant X such that the above-mentioned problems have at least two solutions if X G 0 X a unique positive solution if X X and no solution if X X . We also obtain some bifurcation results of the solutions at X X . Copyright 2007 Tsing-San Hsu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Throughout this article let N m n 2 n 1 2 2N N - 2 for N 3 2 TO for N 2 x y z be the generic point of RN with y G Rm z G Rn. In this article we are concerned with the following eigenvalue problems -Au u X f u h x in o u in H0 O u 0 in o N 2 1.1 x where X 0 0 G M c Rm is a smooth bounded domain S M X Rn D is a smooth bounded domain in RN such that D GG S o S D is an exterior strip domain in RN h x G L2 o n Lq0 o for some q0 N 2 if N 4 q0 2 if N 2 3 h x 0 h x ị 0 and f satisfies the following conditions f1 f G C1 0 to R f 0 0 and f t ị 0 if t 0 f2 there is a positive constant c such that I f t I c t t p forsome1 p 2 - 1 1.1 2 Boundary Value Problems f3 limMt-1 f t 0 f4 there is a number 0 e 0 1 such that 0tf t f t 0 for t 0 1.2 f5 f e C2 0 0 and f t 0 for t 0 15 f e C2 0 0 and f t 0 for t 0 f6 limt- .0- t1-q1 f t C where C is some constant 0 q1 4 N - 2 if N 3 q1 0